Page 99 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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THE FINITE ELEMENT METHOD
The above equation can be written in a compact form as
[K]{T}={f} (3.260)
where
T T
[K] = [B] [D][B]d
+ h[N] [N]ds

S 3

T T T
{f}= G[N] d
− q[N] ds + hT a [N] ds (3.261)

S 2 S 3
Equations 3.260 form the backbone of the calculation method for a finite element analy-
sis of heat conduction problems. It can be easily noted that when there is no heat generation
within an element (G = 0), the corresponding term disappears. Similarly, for an insulated
boundary (i.e., q = 0or h = 0) the corresponding term again disappears. Thus, for an insu-
lated boundary, we do not have to specify any contribution, but leave it unattended. In this
respect, this is a great deal more convenient as compared to the finite difference method,
where nodal equations have to be written for insulated boundaries.

3.4.2 The Galerkin method
The method requires that the following expression be satisfied:

w k L(T)d
= 0 (3.262)

where the weight w k is replaced by the shape functions at nodes, N k (x),thatis,
$ ! ! ! %
∂ ∂T ∂ ∂T ∂ ∂T

N k k x + k y + k z + G d
= 0 (3.263)

∂x ∂x ∂y ∂y ∂z ∂z
Integration by parts is often essential when dealing with second-order derivatives. Using
Green’s lemma (see Appendix A), we can rewrite the second derivatives in two parts as
! !
∂ ∂T ∂T ∂N k ∂N m

N k k x d
= N k k x ds − k x {T m }d
(3.264)

∂x ∂x S ∂x
∂x ∂x
where m represents nodes. With the boundary conditions (3.242), we can rewrite
Equation 3.263 as

∂N k ∂N m ∂N k ∂N m ∂N k ∂N m
− k x + k y + k z {T m }d

∂x ∂x ∂y ∂y ∂z ∂z

+ GN k d
− N k qds + hN k N m {T m }ds + hT a N k ds = 0 (3.265)

S S S
Now collecting the coefficients of the nodal variables {T m },weget
[K]{T }= {f} (3.266)
or
[K km ]{T m }= {f k } (3.267)